96 V NORM ED SPACES

(5.2.2) If the series (xn) and (x'n) are convergent and have sums s, s't then
the series (xn 4- x'n) converges to the sum s + sf and the series (lxn) to the
sum Is for any scalar L

Follows at once from the definition and from (5.1.5).

(5.2.3) If (xn) and (x'n) are two series such that x'n = xn except for a finite
number of indices, they are both convergent or both nonconvergent.

For the series (x'n — xn) is convergent, since all its terms are 0 except
for a finite number of indices.

(5.2.4) Let (kn) be a strictly increasing sequence of integers ^0 with &0 = 0;

fcn+i-i
if the series (xn) converges to s, and ifyn = ]T xp, then the series (yn) converges

p = kn

also to s.

n kn +1 - 1

This follows at once from the relation £ yt = £ Xj and from (3.13.10).

PROBLEMS

1. Let (an) be an arbitrary sequence in a normed space E; show that there exists a sequence
. (*„) of points of E such that lim xn = 0, and a strictly increasing sequence (/:„) of

w-+0

integers such that an = x0 -f- Xi H-----+ xkn for every n.

2. Let a be a bijection of N onto itself, and for each n, let <p(ri) be the smallest number of
intervals [a, b] in N such that the union of these intervals is a([0, n])

(a) Suppose 9? is bounded in N. Let (xn) be a convergent series in a normed space E;

00 00

show that the series (*„(„)) is convergent in E and that 2 *«= 2 **<«> •

n = 0 nsaO

(b) Suppose cp is unbounded in N. Define a series (xn) of real numbers which is con-
vergent, but such that the series (x^) is not convergent in R. (Define by induction on k
a strictly increasing sequence (mfc) of integers having the following properties:

(1) If nk is the largest element of o-([0, mk])> then [0, nk] is contained in a([0, mk+i]).

(2) <p(mk)^k + l.

Define then xn for nk <n^nk+i such that xa = 0 except for 2k conveniently chosen
values of n, at each of which xn is alternately equal to I/A: or to — I/A:.)

3. Let (xn) be a convergent series in a normed space E; let cr be a bijection of N onto itself,
and let

r(n)^\ar(n)-~n\ • sup ||*m||. = x0, b = xn and that the broken line L defined by that sequence is con-